The basic model which I will use to assess the impact of pharmaceutical innovation and cancer incidence on age-standardized cancer mortality rates in Mexico is:

$$ \begin{aligned} {\text{MORT}}_{st} = \beta_{k} {\text{CUM}}\_{\text{NCE}}_{s,t - k} + \gamma \,{\text{INCIDENCE}}_{s,t - 1} + \alpha_{s} + \delta_{t} + \varepsilon_{st} , \\ \end{aligned} $$

(1)

where MORT_{
st
} = the age-standardized mortality rate from cancer at site *s* in year *t* (*t* = 2003, 2013); CUM_NCE_{
s,t–k
} = ∑_{
d
} IND_{
ds
} LAUNCHED_{
d,t−k
} = the number of new chemical entities (drugs) to treat cancer at site *s* that had been launched in Mexico by the end of year *t*−*k* (*k* = 0, 3, 6 ,…,18); \( {\text{IND}}_{ds} \begin{array}{ll} {{ = }\; 1 {\text{ if drug }}d{\text{ is used to treat }}\left( {\text{indicated for}} \right){\text{ cancer at site }}s} \\ { = 0{\text{ if drug }}d{\text{ is not used to treat }}\left( {\text{indicated for}} \right){\text{ cancer at site }}s} \\ \end{array} ; \) \( {\text{LAUNCHED}}_{d,t - k} \begin{array}{*{20}c} {{ = }\, 1 {\text{ if drug }}d{\text{ had been launched in Mexico by the end of year }}t - k} \\ {{ = }\,0{\text{ if drug }}d{\text{ had not been launched in Mexico by the end of year }}t - k} \\ \end{array} ; \) INCIDENCE_{
s,t−1} = the age-standardized incidence rate of cancer at site *s* in year *t* − 1; *α*
_{
s
} = a fixed effect for cancer at site *s*; *δ*
_{
t
} = a fixed effect for year *t*.

Inclusion of year and cancer-site fixed effects controls for the overall decline in cancer mortality and for stable between-cancer-site differences in mortality. Negative and significant estimates of *β*
_{
k
} in Eq. (1) would signify that cancer sites for which there was more pharmaceutical innovation had larger declines in mortality, controlling for changes in incidence.

Due to data limitations, the number of new chemical entities is the only cancer-site-specific, time-varying, measure of medical innovation in Eq. (1). Both a patient-level US study and a longitudinal country-level study have shown that controlling for numerous other potential determinants of mortality does not reduce, and may even increase, the estimated effect of pharmaceutical innovation. The study based on patient-level data (Lichtenberg 2013) found that controlling for race, education, family income, insurance coverage, Census region, BMI, smoking, the mean year the person started taking his or her medications, and over 100 medical conditions had virtually no effect on the estimate of the effect of pharmaceutical innovation (the change in drug vintage) on life expectancy. The study based on longitudinal country-level data (Lichtenberg 2014b) found that controlling for ten other potential determinants of longevity change [real per capita income, the unemployment rate, mean years of schooling, the urbanization rate, real per capita health expenditure (public and private), the DPT immunization rate among children ages 12–23 months, HIV prevalence, and tuberculosis incidence] *increased* the coefficient on pharmaceutical innovation by about 32%.

Failure to control for non-pharmaceutical medical innovation (e.g., innovation in diagnostic imaging, surgical procedures, and medical devices) is also unlikely to bias estimates of the effect of pharmaceutical innovation on premature mortality, for two reasons. First, more than half of US funding for biomedical research came from pharmaceutical and biotechnology firms (Dorsey et al. 2010). Much of the rest came from the federal government (i.e., the NIH), and new drugs often build on upstream government research (Sampat and Lichtenberg 2011). The National Cancer Institute (2016a, b) says that it “has played an active role in the development of drugs for cancer treatment for 50 years… [and] that approximately one half of the chemotherapeutic drugs currently used by oncologists for cancer treatment were discovered and/or developed” at the National Cancer Institute. Second, the previous research based on US data (Lichtenberg 2014a, c) indicates that non-pharmaceutical medical innovation is not positively correlated across diseases with pharmaceutical innovation. However, while non-pharmaceutical medical innovation may not be correlated with pharmaceutical innovation across diseases in the US, this need not hold for Mexico.

The measure of pharmaceutical innovation in Eq. (1)—the number of chemical substances previously registered to treat cancer at site *s*—is not the theoretically ideal measure. Mortality is presumably more strongly related to the drugs *actually* used to treat cancer than it is to the drugs that *could be* used to treat cancer. A preferable measure is the mean *vintage* of drugs used to treat cancer at site *s* in year *t*, defined as VINTAGE_{
st
} = ∑_{
d
} Q_{
dst
} LAUNCH_YEAR_{
d
}/∑_{
d
} Q_{
dst
}, where Q_{
dst
} = the quantity of drug *d* used to treat cancer at site *s* in year *t*, and LAUNCH_YEAR_{
d
} = the world launch year of drug *d*.^{Footnote 7} Unfortunately, measurement of VINTAGE_{
st
} is infeasible: even though data on the total quantity of each drug in each year (Q_{
d.t
} = Σ_{
s
} Q_{
dst
}) are available, many drugs are used to treat multiple diseases. There is no way to determine the quantity of drug *d used to treat cancer at site s* in year *t*.^{Footnote 8} However, Lichtenberg (2014c) showed that in France, there is a highly significant positive correlation across *drug classes* between changes in the (quantity-weighted) vintage of drugs and changes in the number of chemical substances previously registered within the drug class.

In principle, it could be desirable to control for the length of time between the ‘world launch’ of cancer drugs and their ‘Mexico launch’.^{Footnote 9} According to Solow’s vintage hypothesis, later, vintage goods (e.g., drugs whose world launch years were later) are likely to be of higher quality than earlier vintage goods. Holding constant the Mexican launch year of a drug, the shorter the lag from world launch year to Mexican launch year, the later the world launch year of the drug, and (according to the vintage hypothesis), the higher the drug’s quality. However, controlling for the length of time between the ‘world launch’ of cancer drugs and their ‘Mexico launch’ is problematic. I can compute the mean lag between the world launch year and the Mexican launch year for cancer sites and years in which at least one drug had been launched in Mexico. However, I cannot compute the mean lag for cancer sites and years in which no drugs had been launched in Mexico. As shown in Table 2, 8 cancer sites had 0 drug launches by 1995; 4 cancer sites had 0 drugs launches by 2004. Controlling for the length of time between the ‘world launch’ of cancer drugs and their ‘Mexico launch’ would require excluding those observations.

In Eq. (1), mortality from cancer at site *s* in year *t* depends on the number of new chemical entities (drugs) to treat cancer at site *s* that had been launched in Mexico by the end of year *t* − *k*, i.e., there is a lag of *k* years. Equation (1) will be estimated for different values of *k*: *k* = 0, 3, 6,…,18. A separate model is estimated for each value of *k*, rather than including multiple values (CUM_NCE_{
s,t
}, CUM_NCE_{
s,t−3}, CUM_NCE_{
s,t−6},…) in a single model, because CUM_NCE is highly serially correlated (by construction), which would result in extremely high multicollinearity if multiple values were included. One would expect there to be a substantial lag, because new drugs diffuse gradually—they will not be used widely until years after registration. Data from the IMS Health MIDAS database can be used to provide evidence about the process of diffusion of new medicines. I used data from that source linked to data on Mexican drug launch dates (described below) to estimate the following model:

$$ { \ln }\left( {{\text{N}}\_{\text{RX}}_{dy} } \right) \, = \rho_{d} + \pi_{y} + \varepsilon_{dy} , $$

(2)

where N_RX_{
dy
}= the number of standard units of cancer drug *d* sold in Mexico per thousand population *y* years after it was launched (*y* = 0–3, 4–7, 8–11, 12–15, 16–19 years); *ρ*
_{
d
} = a fixed effect for drug *d*; *π*
_{
y
} = a fixed effect for age *y*.

The expression \( \exp \left( {\pi_{y} - \pi_{16 - 19} } \right) \) is a “relative utilization index”: it is the mean ratio of the annual number of standard units of a cancer drug sold per thousand population *y* years after it was first launched in Mexico to the annual number of standard units of the same drug sold per thousand population 16–19 years after it was first launched in Mexico.

Using annual data on the number of standard units of cancer drugs sold in Mexico during the period 1999–2010, I estimated Eq. (2). Estimates of the “relative utilization index” are shown in Fig. 3. These estimates indicate that utilization of a drug is strongly positively related to how long the drug has been on the market. On average, a drug is used 50 times as often per year 16–19 years post-launch as it is 0–3 year post-launch. Utilization appears to rise especially rapidly after year 11, when drug patents tend to expire and generics enter the market.

The effect of a drug’s launch on mortality is likely to depend on both the *quality* and the *quantity* of the drug. Indeed, it is likely to depend on the *interaction* between quality and quantity: a quality improvement will have a greater impact on mortality if drug utilization (quantity) is high. Although newer drugs tend to be of higher quality than older drugs (see Lichtenberg 2014d), the relative quantity of very new drugs is quite low, so the impact on mortality of very new drugs is lower than the impact of older drugs.

In principle, mortality in year *t* should depend on a distributed lag of incidence, i.e., on INCIDENCE_{
s,t
}, INCIDENCE_{
s,t−1}, INCIDENCE_{
s,t−2}, INCIDENCE_{
s,t−3}… Unfortunately, data on incidence by cancer site are available for only 2 years (2002 and 2012); this is why INCIDENCE_{
s,t−2} is the only incidence variable included in Eq. (1). The limited availability of incidence data also means that we can only use mortality data for 2 years (2004 and 2014). Writing the model for each of these years:

$$ \begin{aligned} {\text{MORT}}_{s,2003} = \beta_{k} {\text{CUM}}\_{\text{NCE}}_{s, 200 3- k} + \gamma {\text{INCIDENCE}}_{s, 200 2} + \alpha_{s} + \delta_{ 200 3} + \varepsilon_{s,2003} , \\ \end{aligned} $$

(3)

$$ \begin{aligned} {\text{MORT}}_{s, 20 1 3} = \beta_{k} {\text{CUM}}\_{\text{NCE}}_{s,2013 - k} \ + \gamma \,{\text{INCIDENCE}}_{s,2012} + \alpha_{s} + \delta_{ 20 1 3} + \varepsilon_{s,2013} . \\ \end{aligned} $$

(4)

Subtracting (3) from (4),

$$ \begin{aligned} \left( {{\text{MORT}}_{s, 20 1 3} - {\text{ MORT}}_{s, 200 3} } \right) \, & = \beta_{k} \left( {{\text{CUM}}\_{\text{NCE}}_{s, 20 1 3- k} - {\text{ CUM}}\_{\text{NCE}}_{s, 200 3- k} } \right) \\ & \quad + \gamma \left( {{\text{INCIDENCE}}_{s, 20 1 2} - {\text{ INCIDENCE}}_{s,2002} } \right) \\ \, & \quad + \, (\delta_{ 20 1 3} - \delta_{ 200 3} ) \, + \, (\varepsilon_{s,2013} - \varepsilon_{s,2003} ). \\ \end{aligned} $$

(5)

Equation (5) may be rewritten as follows:

$$ \begin{aligned} \Delta {\text{MORT}}_{s} & = \beta_{k} \Delta {\text{CUM}}\_{\text{NCE}}\_k_{s} \\ & \quad + \gamma \,\Delta {\text{INCIDENCE}}\_ 1_{s} + \delta^{\prime} \, + \varepsilon_{s}^{\prime } , \\ \end{aligned} $$

(6)

where ∆MORT_{
s
} = MORT_{
s,2013} − MORT_{
s,2003} = the 2003–2013 change in the age-standardized mortality rate from cancer at site *s*; ∆CUM_NCE_*k*
_{
s
} = CUM_NCE_{
s,2013k
} − CUM_NCE_{
s,2003−k
} = the number of drugs for cancer at site *s* launched between year 2003 − *k* and 2013 − *k*; ∆INCIDENCE_1_{
s
} = INCIDENCE_{
s,2012} − INCIDENCE_{
s,2002} = the 2002–2012 change in the age-standardized incidence rate of cancer at site *s*; *δ*′ = *δ*
_{2013} − *δ*
_{2003}.

Equation (6) indicates that the 2003–2013 change in the age-standardized mortality rate depends on two variables: the number of drugs launched between year 2003 − k and 2013 − k, and the 2002–2012 change in the age-standardized incidence rate.

For estimates of *β*
_{
k
} from Eqs. (1) and (6) to be consistent estimates of the effect of drug launches on mortality, the “parallel trends” assumption needs to be satisfied. A simple way to test the validity of this assumption is to estimate a version of Eq. (6) that includes a control for the trend in mortality in the ‘pre-period’, e.g., the period 1993–2003. Therefore, I will estimate the following model:

$$ \begin{aligned} \Delta {\text{MORT}}_{s} & = \beta_{k} \Delta {\text{CUM}}\_{\text{NCE}}\_k_{s} + \gamma \,\Delta {\text{INCIDENCE}}\_ 1_{s} \\ & \quad + \pi \, \Delta {\text{MORT}}\_{\text{PRE}}_{s} + \delta^{\prime} \, + \varepsilon_{s}^{\prime } , \\ \end{aligned} $$

(7)

where ∆MORT_PRE_{
s
} = MORT_{
s,2003} − MORT_{
s,1993} = the 1993–2003 change in the age-standardized mortality rate from cancer at site *s*.